Apollonius <Pergaeus>; Lawson, John, The two books of Apollonius Pergaeus, concerning tangencies, as they have been restored by Franciscus Vieta and Marinus Ghetaldus : with a supplement to which is now added, a second supplement, being Mons. Fermat's Treatise on spherical tangencies

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        <div xml:id="echoid-div11" type="section" level="1" n="11">
          <pb o="(4)" file="0016" n="16"/>
          <p>
            <s xml:id="echoid-s189" xml:space="preserve">
              <emph style="sc">Limitation</emph>
            . </s>
            <s xml:id="echoid-s190" xml:space="preserve">2Z muſt not be given leſs than the perpendicular let fall from
              <lb/>
            the given point A upon the given line BC.</s>
            <s xml:id="echoid-s191" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s192" xml:space="preserve">
              <emph style="sc">From</emph>
            the point A let AD be drawn perpendicular to BC, and in this perpen-
              <lb/>
            dicular take DE equal to the given line Z: </s>
            <s xml:id="echoid-s193" xml:space="preserve">and through E draw EF parallel to
              <lb/>
            BC, and from A upon this line EF ſet off AF equal to Z, which may be done,
              <lb/>
            for by the Limitation Z is not leſs than AE: </s>
            <s xml:id="echoid-s194" xml:space="preserve">then with center F and diſtance
              <lb/>
            FA deſcribe a circle, and I ſay it will touch the line BC: </s>
            <s xml:id="echoid-s195" xml:space="preserve">for through F drawing
              <lb/>
            FG parallel to AD, FGDE will be a Parallelogram, and FG will be equal to
              <lb/>
            DE, that is to Z, and at right angles to BC.</s>
            <s xml:id="echoid-s196" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div12" type="section" level="1" n="12">
          <head xml:id="echoid-head17" xml:space="preserve">PROBLEM V.</head>
          <p>
            <s xml:id="echoid-s197" xml:space="preserve">
              <emph style="sc">Having</emph>
            a given point A, and alſo a given circle whoſe center is B, it is re-
              <lb/>
            quired to draw a circle whoſe Radius ſhall be equal to a given line Z, which ſhall
              <lb/>
            paſs through the given point, and alſo touch the given circle.</s>
            <s xml:id="echoid-s198" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s199" xml:space="preserve">
              <emph style="sc">This</emph>
            Problem has three Caſes, each of which is ſubject to a Limitation.</s>
            <s xml:id="echoid-s200" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s201" xml:space="preserve">
              <emph style="sc">Case</emph>
            Iſt. </s>
            <s xml:id="echoid-s202" xml:space="preserve">Let the circle to be doſcribed be required to be touched outwardly
              <lb/>
            by the given circle.</s>
            <s xml:id="echoid-s203" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s204" xml:space="preserve">
              <emph style="sc">Limitation</emph>
            . </s>
            <s xml:id="echoid-s205" xml:space="preserve">Then the Diameter muſt not be given leſs than the ſegment
              <lb/>
            of the right line, joining the given point and the center of the given circle,
              <lb/>
            which is intercepted between the given point and the convex circumference; </s>
            <s xml:id="echoid-s206" xml:space="preserve">viz.
              <lb/>
            </s>
            <s xml:id="echoid-s207" xml:space="preserve">not leſs than AC.</s>
            <s xml:id="echoid-s208" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s209" xml:space="preserve">
              <emph style="sc">Case</emph>
            2d. </s>
            <s xml:id="echoid-s210" xml:space="preserve">Let the circle to be deſcribed be required to be touched inwardly
              <lb/>
            by the given circle.</s>
            <s xml:id="echoid-s211" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s212" xml:space="preserve">
              <emph style="sc">Limitation</emph>
            . </s>
            <s xml:id="echoid-s213" xml:space="preserve">Then the Diameter muſt not be given leſs than the right line
              <lb/>
            which, drawn from the given point through the center of the given circle, is con-
              <lb/>
            tained between the given point and the concave circumſerence; </s>
            <s xml:id="echoid-s214" xml:space="preserve">viz. </s>
            <s xml:id="echoid-s215" xml:space="preserve">not leſs
              <lb/>
            than AC.</s>
            <s xml:id="echoid-s216" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s217" xml:space="preserve">
              <emph style="sc">Case</emph>
            3d. </s>
            <s xml:id="echoid-s218" xml:space="preserve">Let the given point lie in the given circle.</s>
            <s xml:id="echoid-s219" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s220" xml:space="preserve">
              <emph style="sc">Limitation</emph>
            . </s>
            <s xml:id="echoid-s221" xml:space="preserve">Then a diameter of the given circle being drawn through the
              <lb/>
            given point, it is divided into two ſegments by the ſaid point, and the Diameter
              <lb/>
            of the circle required muſt not be given greater than the greater of them, nor
              <lb/>
            leſs than the leſſer; </s>
            <s xml:id="echoid-s222" xml:space="preserve">viz. </s>
            <s xml:id="echoid-s223" xml:space="preserve">not greater than AC, nor leſs than AG.</s>
            <s xml:id="echoid-s224" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div13" type="section" level="1" n="13">
          <head xml:id="echoid-head18" xml:space="preserve">
            <emph style="sc">The general</emph>
            <emph style="sc">Solution</emph>
          .</head>
          <p>
            <s xml:id="echoid-s225" xml:space="preserve">
              <emph style="sc">Let</emph>
            A and B be joined, and in the line AB take CF equal to Z, and then
              <lb/>
            with center A and diſtance Z, let an arc be drawn, and with center B, and
              <lb/>
            diſtance BF let another be drawn, which by the Limitations will neceſſarily
              <lb/>
            either touch or cut the former; </s>
            <s xml:id="echoid-s226" xml:space="preserve">let the point of their concourſe be D; </s>
            <s xml:id="echoid-s227" xml:space="preserve">then with
              <lb/>
            D center and DA diſtance let a circle be drawn, and I ſay it will touch the
              <lb/>
            given circle whoſe center is B: </s>
            <s xml:id="echoid-s228" xml:space="preserve">for DB being drawn meeting the circumference
              <lb/>
            of the circle whoſe center is B in E, BC is equal to BE, and hence CF equals
              <lb/>
            ED, and they are both equal to the given line Z.</s>
            <s xml:id="echoid-s229" xml:space="preserve"/>
          </p>
        </div>
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